Subsections

• Related Structures
• Numerical Invariants
• Polynomials for Finite Fields
• Ring Predicates and Booleans
• Roots

Category(F) : FldFin -> Cat

Parent(F) : FldFin -> PowerStructure

Centre(F) : FldFin -> FldFin

PrimeRing(F) : FldFin -> FldFin

PrimeField(F) : FldFin -> FldFin

FieldOfFractions(F) : FldFin -> FldFin

Given F=GF(q), create the finite additive abelian group A of order q=p^r that is the direct sum of r copies of the cyclic group of order p, together with a map from the field F to A that gives an isomorphism.

UnitGroup(F) : FldFin -> GrpAb, Map

Given F=GF(q), create the multiplicative group of R as an abelian group. This returns the (additive) cyclic group A of order q - 1, together with a (partial) map from F - 0 to A, sending a primitive element of F to 1.

Create the enumerated set consisting of the elements of finite field F.

Given a finite field F that is an extension of degree n of E, define the natural isomorphism between F and the n-dimensional vector space E^n. The function returns two values:

• A vector space V isomorphic to E^n;
• The isomorphism phi : F -> V.

The basis of V is chosen to correspond with the power basis alpha^0, alpha^1, ..., alpha^(n - 1) of F, where alpha is the generator returned by Generator(F, E), so that V=E.1 x E.alpha x ... x E.alpha^(n - 1) and phi : alpha^i -> e_(i + 1), (for i = 0, ..., n - 1), where e_i is the basis vector of V having all components zero, except the i-th, which is one.

VectorSpace(F, E, B) : FldFin, FldFin, [ FldFinElt ] -> ModTupFld, Map

Given a finite field F that is an extension of degree n of E, define the isomorphism between F and the n-dimensional vector space E^n defined by the basis B for F over E. The function returns two values:

• A vector space V isomorphic to E^n;
• The isomorphism phi : F -> V.

The basis of V is chosen to correspond with the basis B=beta_1, beta_2, ..., beta_(n) of F over E, as specified by the user, so that V=E.beta_1 x E.beta_2 x ... x E.beta_(n). phi : beta_i -> e_(i), (for i = 1, ..., n), where e_i is the basis vector of V having all components zero, except the i-th, which is one.

Let F be a finite field that is an extension of degree n of E. The function returns two values:

• A matrix algebra A of degree n, such that A is isomorphic to F;
• An isomorphism phi : F -> A.

The matrix algebra A will be the subalgebra of the full algebra of n x n matrices over E generated by the companion matrix C of the defining polynomial of F over E. The generator Generator(F, E) of F over E is thus mapped to C.

MatrixAlgebra(A, E) : AlgMat, FldFin -> AlgMat, Map

Let F be a finite field. Let A be a matrix algebra over F, and E be a subfield of F. The function returns two values:

• A matrix algebra N over E isomorphic to A, obtained from A by expanding each component of an element of A into the block matrix associated with it;
• An E-isomorphism phi : A -> N.

N is A considered as an E-matrix algebra.

> F7 := FiniteField(7);
> F49<w> := ext< F7 | 2 >;
> F<z> := ext< F49 | 2 >;
> v2, i2 := VectorSpace(F, F49);
> v2;
Full Vector space of degree 2 over GF(7^2)
> i2(z^12);
(   w w^28)
> v4, i4 := VectorSpace(F, PrimeField(F));
> v4;
Full Vector space of degree 4 over GF(7)
> i4(z^12);
(5 3 6 4)

Characteristic(F) : FldFin -> FldFinElt

# F : FldFin -> FldFinElt

The absolute degree of F, that is, the degree over its prime subfield.

Degree(F, E) : FldFin, FldFin -> RngIntElt

Given a finite field F that has been constructed as an extension of a field E, return the degree of F over E.

Given a finite field F that has been constructed as an extension of a field E, return the polynomial with coefficients in E that was used to define F as an extension of E. This is the minimum polynomial of F.1.

DefiningPolynomial(F, E) : FldFin -> RngPolElt

Given a finite field F and a subfield E, return the polynomial with coefficients in E used to define F as an extension of E. This is the same as the minimum polynomial of the generator Generator(F, E) over E.

Given a finite field F and a positive integer m > 1, construct a polynomial P of degree m that is irreducible over F.

Given a finite field F and a positive integer m > 1, construct a polynomial P of degree m that is primitive over F. Thus, P is irreducible over F, and it has a primitive root of the degree m extension field of F as a root.

Given a finite field F and a positive integer m > 1, construct the set of all monic polynomials of degree m that are irreducible over F.

Given a prime p and an exponent n >= 1, return the Conway polynomial of degree n over GF(p). The Conway polynomial is defined in the introduction. Note that this polynomial is read in from a table containing Conway polynomials for a limited range of p, n only.

Given a prime p and an exponent n>1, return true and the Conway polynomial if it is known for the field GF(p), false otherwise.

IsCommutative(F) : FldFin -> BoolElt

IsUnitary(F) : FldFin -> BoolElt

IsFinite(F) : FldFin -> BoolElt

IsOrdered(F) : FldFin -> BoolElt

IsField(F) : FldFin -> BoolElt

IsEuclideanDomain(F) : FldFin -> BoolElt

IsPID(F) : FldFin -> BoolElt

IsUFD(F) : FldFin -> BoolElt

IsDivisionRing(F) : FldFin -> BoolElt

IsEuclideanRing(F) : FldFin -> BoolElt

IsPrincipalIdealRing(F) : FldFin -> BoolElt

IsDomain(F) : FldFin -> BoolElt

F eq G : FldFin, Rng -> BoolElt

F ne G : FldFin, Rng -> BoolElt

Given a finite field F, this function returns true if F is defined over the prime field using the Conway polynomial, false otherwise.

Given a polynomial f over a finite field F, this function finds all roots of f in F, and returns a sequence of tuples (pairs), each consisting of a root of f in F and its multiplicity.

A primitive n-th root of unity in the smallest possible extension field of K.